i need to find $$\int_{\Gamma}z^i\,dz$$ along the upper semi-circle of radius $1$ and $-\pi<Arg(z)<\pi$.
I paramatrized $\Gamma$ as $e^{i\theta}, 0\leq\theta\leq\pi$: $$\int_0^{\pi}ie^{i\theta}\left(e^{i\theta}\right)^id\theta=\int_0^{\pi}ie^{i\theta}e^{-\theta}d\theta=i\int_0^{\pi}e^{(i-1)\theta}d\theta=\frac{1-i}{2}e^{(1-i)\theta}\Big|_0^{\pi}=\frac{1-i}{2}(-e^{-\pi}-1)$$ Is this ok? Because my book dos this $$z^i=e^{iLog(z)}$$ and plugs para. for $\Gamma$ into $e^{iLog(z)}$. But answers are the same, why does the book do this long way?
For complex numbers $a$ and $b$ the power $a^{b}$ has to be defined carefully. Here, the author is defining it as $e^{b Log (a)}$ where Log is the principal branch of logarithm. Your claim that $(e^{i\theta})^{i} =e^{-\theta}$ requies a proof and the proof is by going to the definition. In other words the author is making your computation rigorous.